Question: $ D = \left[\begin{array}{rrr}0 & -2 & 0 \\ -1 & 4 & 3\end{array}\right]$ $ E = \left[\begin{array}{rr}1 & 2 \\ -2 & 2 \\ 0 & 3\end{array}\right]$ What is $ D E$ ?
Answer: Because $ D$ has dimensions $(2\times3)$ and $ E$ has dimensions $(3\times2)$ , the answer matrix will have dimensions $(2\times2)$ $ D E = \left[\begin{array}{rrr}{0} & {-2} & {0} \\ {-1} & {4} & {3}\end{array}\right] \left[\begin{array}{rr}{1} & \color{#DF0030}{2} \\ {-2} & \color{#DF0030}{2} \\ {0} & \color{#DF0030}{3}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ D$ , with the corresponding elements in column $j$ of the second matrix, $ E$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ D$ with the first element in ${\text{column }1}$ of $ E$ , then multiply the second element in ${\text{row }1}$ of $ D$ with the second element in ${\text{column }1}$ of $ E$ , and so on. Add the products together. $ \left[\begin{array}{rr}{0}\cdot{1}+{-2}\cdot{-2}+{0}\cdot{0} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ D$ with the corresponding elements in ${\text{column }1}$ of $ E$ and add the products together. $ \left[\begin{array}{rr}{0}\cdot{1}+{-2}\cdot{-2}+{0}\cdot{0} & ? \\ {-1}\cdot{1}+{4}\cdot{-2}+{3}\cdot{0} & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ D$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ E$ and add the products together. $ \left[\begin{array}{rr}{0}\cdot{1}+{-2}\cdot{-2}+{0}\cdot{0} & {0}\cdot\color{#DF0030}{2}+{-2}\cdot\color{#DF0030}{2}+{0}\cdot\color{#DF0030}{3} \\ {-1}\cdot{1}+{4}\cdot{-2}+{3}\cdot{0} & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{0}\cdot{1}+{-2}\cdot{-2}+{0}\cdot{0} & {0}\cdot\color{#DF0030}{2}+{-2}\cdot\color{#DF0030}{2}+{0}\cdot\color{#DF0030}{3} \\ {-1}\cdot{1}+{4}\cdot{-2}+{3}\cdot{0} & {-1}\cdot\color{#DF0030}{2}+{4}\cdot\color{#DF0030}{2}+{3}\cdot\color{#DF0030}{3}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}4 & -4 \\ -9 & 15\end{array}\right] $